### Abstract

The Bonami-Beckner hypercontractive inequality is a powerful tool in Fourier analysis of real-valued functions on the Boolean cube. In this paper we present a version of this inequality for matrix-valued functions on the Boolean cube. Its proof is based on a powerful inequality by Ball, Carlen, and Lieb. We also present a number of applications. First, we analyze maps that encode n classical bits into m qubits, in such a way that each set of k bits can be recovered with some probability by an appropriate measurement on the quantum encoding; we show that if m < 0.7n, then the success probability is exponentially small in k. This result may be viewed as a direct product version of Nayak's quantum random access code bound. It in turn implies strong direct product theorems for the one-way quantum communication complexity of Disjointness and other problems. Second, we prove that error-correcting codes that are locally decodable with 2 queries require length exponential in the length of the encoded string. This gives what is arguably the first "non-quantum" proof of a result originally derived by Kerenidis and de Wolf using quantum information theory.

Original language | English (US) |
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Title of host publication | Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008 |

Pages | 477-486 |

Number of pages | 10 |

DOIs | |

State | Published - 2008 |

Event | 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008 - Philadelphia, PA, United States Duration: Oct 25 2008 → Oct 28 2008 |

### Other

Other | 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008 |
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Country | United States |

City | Philadelphia, PA |

Period | 10/25/08 → 10/28/08 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)

### Cite this

*Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008*(pp. 477-486). [4690981] https://doi.org/10.1109/FOCS.2008.45

**A hypercontractive inequality for matrix-valued functions with applications to quantum computing and LDCs.** / Ben-Aroya, Avraham; Regev, Oded; De Wolf, Ronald.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008.*, 4690981, pp. 477-486, 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, Philadelphia, PA, United States, 10/25/08. https://doi.org/10.1109/FOCS.2008.45

}

TY - GEN

T1 - A hypercontractive inequality for matrix-valued functions with applications to quantum computing and LDCs

AU - Ben-Aroya, Avraham

AU - Regev, Oded

AU - De Wolf, Ronald

PY - 2008

Y1 - 2008

N2 - The Bonami-Beckner hypercontractive inequality is a powerful tool in Fourier analysis of real-valued functions on the Boolean cube. In this paper we present a version of this inequality for matrix-valued functions on the Boolean cube. Its proof is based on a powerful inequality by Ball, Carlen, and Lieb. We also present a number of applications. First, we analyze maps that encode n classical bits into m qubits, in such a way that each set of k bits can be recovered with some probability by an appropriate measurement on the quantum encoding; we show that if m < 0.7n, then the success probability is exponentially small in k. This result may be viewed as a direct product version of Nayak's quantum random access code bound. It in turn implies strong direct product theorems for the one-way quantum communication complexity of Disjointness and other problems. Second, we prove that error-correcting codes that are locally decodable with 2 queries require length exponential in the length of the encoded string. This gives what is arguably the first "non-quantum" proof of a result originally derived by Kerenidis and de Wolf using quantum information theory.

AB - The Bonami-Beckner hypercontractive inequality is a powerful tool in Fourier analysis of real-valued functions on the Boolean cube. In this paper we present a version of this inequality for matrix-valued functions on the Boolean cube. Its proof is based on a powerful inequality by Ball, Carlen, and Lieb. We also present a number of applications. First, we analyze maps that encode n classical bits into m qubits, in such a way that each set of k bits can be recovered with some probability by an appropriate measurement on the quantum encoding; we show that if m < 0.7n, then the success probability is exponentially small in k. This result may be viewed as a direct product version of Nayak's quantum random access code bound. It in turn implies strong direct product theorems for the one-way quantum communication complexity of Disjointness and other problems. Second, we prove that error-correcting codes that are locally decodable with 2 queries require length exponential in the length of the encoded string. This gives what is arguably the first "non-quantum" proof of a result originally derived by Kerenidis and de Wolf using quantum information theory.

UR - http://www.scopus.com/inward/record.url?scp=57949116863&partnerID=8YFLogxK

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U2 - 10.1109/FOCS.2008.45

DO - 10.1109/FOCS.2008.45

M3 - Conference contribution

AN - SCOPUS:57949116863

SN - 9780769534367

SP - 477

EP - 486

BT - Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008

ER -